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Data-driven model order reduction for wave propagation in materials with nonlinearities or damage

Saddam Hijazi, Nikiema Fulgence, Hannah Burmester, Natalie Rauter, Carmen Gräßle

TL;DR

This work investigates how to accelerate wave propagation simulations in nonlinear or damaged materials through model order reduction, blending intrusive POD-Galerkin and non-intrusive data-driven methods (DMD, OpInf). It analyzes three numerical scenarios—a parameterized wave equation with damage, GUW propagation in a damaged fiber metal laminate, and a nonlinear Neo-Hookean aluminum plate—to benchmark MOR techniques, including a data-scaling enhancement for OpInf and a forces-informed variant. Key findings show POD-Galerkin provides high accuracy at modest reduced dimensions, while DMD/mrDMD capture localized or nonlinear dynamics better in certain regimes; OpInf can match or surpass these in some cases but may require scaling and careful regularization. The results advance SHM/NDE by enabling fast, accurate reduced-order models capable of informing damage identification and real-time monitoring in complex materials.

Abstract

In this work, we consider wave propagation in materials characterized by nonlinear properties or damage. To accelerate the simulations of the resulting high-dimensional problems, we apply model order reduction methods. Depending on the knowledge of the underlying equations and the availability of their discrete operators, intrusive methods (here projection-based approaches based on proper orthogonal decomposition (POD)) or non-instrusive methods (here data-driven approaches including dynamic mode decomposition (DMD) and operator inference (OpInf)) can be used. We recall the theoretical foundations of the methods and apply them to the problem of wave propagation. In three different numerical examples, we evaluate the performance of the reduction techniques.

Data-driven model order reduction for wave propagation in materials with nonlinearities or damage

TL;DR

This work investigates how to accelerate wave propagation simulations in nonlinear or damaged materials through model order reduction, blending intrusive POD-Galerkin and non-intrusive data-driven methods (DMD, OpInf). It analyzes three numerical scenarios—a parameterized wave equation with damage, GUW propagation in a damaged fiber metal laminate, and a nonlinear Neo-Hookean aluminum plate—to benchmark MOR techniques, including a data-scaling enhancement for OpInf and a forces-informed variant. Key findings show POD-Galerkin provides high accuracy at modest reduced dimensions, while DMD/mrDMD capture localized or nonlinear dynamics better in certain regimes; OpInf can match or surpass these in some cases but may require scaling and careful regularization. The results advance SHM/NDE by enabling fast, accurate reduced-order models capable of informing damage identification and real-time monitoring in complex materials.

Abstract

In this work, we consider wave propagation in materials characterized by nonlinear properties or damage. To accelerate the simulations of the resulting high-dimensional problems, we apply model order reduction methods. Depending on the knowledge of the underlying equations and the availability of their discrete operators, intrusive methods (here projection-based approaches based on proper orthogonal decomposition (POD)) or non-instrusive methods (here data-driven approaches including dynamic mode decomposition (DMD) and operator inference (OpInf)) can be used. We recall the theoretical foundations of the methods and apply them to the problem of wave propagation. In three different numerical examples, we evaluate the performance of the reduction techniques.

Paper Structure

This paper contains 19 sections, 62 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model description
  3. Wave equation
  4. Mechanical system resulting from GUW propagation
  5. Nonlinear hyperelastic model
  6. Discrete second order mathematical model
  7. Model order reduction techniques
  8. Snapshot data
  9. The operator inference approach: motivation and data preparation
  10. The intrusive POD-Galerkin ROM for second order mechanical systems
  11. The operator inference approach for second order mechanical systems
  12. Scaling the data for the operator inference approach
  13. Dynamic mode decomposition
  14. Numerical examples
  15. Numerical results for the wave equation
  16. ...and 4 more sections

Figures (17)

  • Figure 1: The solution of the FOM for the wave equation at times $t = 0, 2.5,5.$
  • Figure 2: Decay of the singular values of the snapshot matrix $\mathbf{U}$ (left) and cumulative energy $c_r$ (right) for the wave equation.
  • Figure 3: The full order displacement versus the reduced order approximation at the sensor location for the reduced dimensions $r = 40$ (left) and $r = 80$ (right) for the wave equation, respectively.
  • Figure 4: Relative $L^2$-error $\epsilon_s$ at the sensor location as a function of the reduced dimension $r$ (left) and relative $L^2$-error $\epsilon_u(t)$ over the entire spatial domain as a function of time for the reduced dimension $r=40$ (right) for reduced order solutions computed using POD, DMD and OpInf for the wave equation. The values are in percentages.
  • Figure 5: The POD reduced order solution of dimension $r = 40$ (top) and the difference between the full order and POD reduced order solution (bottom) for the wave equation at times $t = 0, 2.5$ and $t = 5.$
  • ...and 12 more figures

Theorems & Definitions (3)

  • Remark 2.1: Damping
  • Remark 3.1: Hyper reduction
  • Remark 3.2: Operator inference for a nonlinear model